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10.5E: Constant Coefficient Homogeneous Systems II (Exercises)

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Q10.5.1

In Exercises 10.5.1-10.5.12 find the general solution.

1. y=[3417]y

2. y=[0112]y

3. y=[74111]y

4. y=[3111]y

5. y=[41238]y

6. y=[10942]y

7. y=[1316911]y

8. y=[021461042]y

9. y=13[113443210]y

10. y=[111202131]y

11. y=[422231213]y

12. y=[653213211]y

Q10.5.2

In Exercises 10.5.13-10.5.23 solve the initial value problem.

13. y=[11823]y,y(0)=[62]

14. y=[159169]y,y(0)=[58]

15. y=[3417]y,y(0)=[23]

16. y=[724617]y,y(0)=[31]

17. y=[7331]y,y(0)=[02]

18. y=[110112111]y,y(0)=[657]

19. y=[221221332]y,y(0)=[620]

20. y=[744101956]y,y(0)=[691]

21. y=[141361323]y,y(0)=[213]

22. y=[484313119]y,y(0)=[413]

23. y=[51117113408]y,y(0)=[022]

Q10.5.3

The coefficient matrices in Exercises 10.5.24-10.5.32 have eigenvalues of multiplicity 3. Find the general solution.

24. y=[511193224]y

25. y=[11012223216]y

26. y=[644211231]y

27. y=[022153111]y

28. y=[21210224112248]y

29. y=[1128194161]y

30. y=[401131102]y

31. y=[334458235]y

32. y=[310110112]y

Q10.5.4

33. Under the assumptions of Theorem 10.5.1, suppose u and ˆu are vectors such that

(Aλ1I)u=xand (Aλ1I)ˆu=x,

and let

y2=ueλ1t+xteλ1tand ˆy2=ˆueλ1t+xteλ1t.

Show that y2ˆy2 is a scalar multiple of y1=xeλ1t.

34. Under the assumptions of Theorem 10.5.2, let

y1=xeλ1t,y2=ueλ1t+xteλ1t, and y3=veλ1t+uteλ1t+xt2eλ1t2.

Complete the proof of Theorem 10.5.2 by showing that y3 is a solution of y=Ay and that {y1,y2,y3} is linearly independent.

35. Suppose the matrix A=[a11a12a21a22] has a repeated eigenvalue λ1 and the associated eigenspace is one-dimensional. Let x be a λ1-eigenvector of A. Show that if (Aλ1I)u1=x and (Aλ1I)u2=x, then u2u1 is parallel to x. Conclude from this that all vectors u such that (Aλ1I)u=x define the same positive and negative half-planes with respect to the line L through the origin parallel to x.

Q10.5.5

In Exercises 10.5.36-10.5.45 plot trajectories of the given system.

36. y=[3141]y

37. y=[2110]y

38. y=[1335]y

39. y=[5331]y

40. y=[2334]y

41. y=[4332]y

42. y=[0112]y

43. y=[0112]y

44. y=[2110]y

45. y=[0414]y


This page titled 10.5E: Constant Coefficient Homogeneous Systems II (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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